Part of Advances in Neural Information Processing Systems 37 (NeurIPS 2024) Main Conference Track
Hassan Ashtiani, Mahbod Majid, Shyam Narayanan
We study the problem of learning mixtures of Gaussians with approximate differential privacy. We prove that roughly $kd^2 + k^{1.5} d^{1.75} + k^2 d$ samples suffice to learn a mixture of $k$ arbitrary $d$-dimensional Gaussians up to low total variation distance, with differential privacy. Our work improves over the previous best result (which required roughly $k^2 d^4$ samples) and is provably optimal when $d$ is much larger than $k^2$. Moreover, we give the first optimal bound for privately learning mixtures of $k$ univariate (i.e., $1$-dimensional) Gaussians. Importantly, we show that the sample complexity for learning mixtures of univariate Gaussians is linear in the number of components $k$, whereas the previous best sample complexity was quadratic in $k$. Our algorithms utilize various techniques, including the inverse sensitivity mechanism, sample compression for distributions, and methods for bounding volumes of sumsets.